The steady mixed convection non-axisymmetric (Homann, Z. Angew. Math. Mech., vol. 16, 1936, pp. 153–164) stagnation-point flow over a vertical flat wall placed in a viscous and incompressible fluid is considered. A similarity solution is derived which involves the dimensionless parameters
$\unicode[STIX]{x1D6FE}$, representing the shear-to-strain-rate ratio, and
$\unicode[STIX]{x1D706}$, a mixed convection parameter. Forced convection,
$\unicode[STIX]{x1D706}=0$, is treated first where solutions additional to those given previously by Weidman (J. Fluid Mech., vol. 702, 2012, pp. 460–469) are found arising from singularities as
$\unicode[STIX]{x1D6FE}\rightarrow \pm 1$. Numerical solutions are obtained for representative values of both
$\unicode[STIX]{x1D6FE}$ and
$\unicode[STIX]{x1D706}$. Critical values
$\unicode[STIX]{x1D706}_{c}$ of
$\unicode[STIX]{x1D706}$ are seen in opposing flow and these are treated in detail. Asymptotic results for large
$\unicode[STIX]{x1D706}$ and
$\unicode[STIX]{x1D6FE}$ are derived.